#8105940

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#53232	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(18,23)(22,28)(24,30)(29,32)(35,41)(38,44)(42,47)(45,48);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(19,25)(22,28)(26,31)(29,32)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(1,5,6,4)(2,8,10,9)(3,12,13,14)(7,16,11,15)(18,23)(22,28)(24,30)(29,32)(34,40)(35,41)(39,46)(42,47);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013136);